Photoelastic Determination of Boundary Condition for Finite Element Analysis Part 2

EXPERIMENTAL VERIFICATION OF THE PROPOSED METHOD

A simple static problem is analyzed to verify the proposed method. A perforated plate made of epoxy resin, 228 mm in height, 50 mm in width and 3 mm in thickness, having a hole of diameter of 10 mm, is subjected to the tensile load of P = 398 N as shown in Fig. 3. The material fringe value f of the material is determined as 11.48 kN/m. The specimen is placed in a circular polariscope with the quarter-wave plates matched for the wavelength of 560 nm. Three monochromatic lights of wavelengths of 500 nm, 550 nm, and 600 nm emitted from a halogen lamp with interference filters are used as the light source in order to apply the absolute phase analysis method with tricolor images [28]. The phase-stepped photoelastic fringes are collected by a monochromatic CCD camera with the resolution of 640 x 480 pixels and 256 gray levels. Then, the fringe pattern is analyzed, the ambiguity of isochromatic phase is corrected and the phase unwrapping is performed by the method proposed previously [28].

Figure 4(a) shows an example of the photoelastic fringe pattern around the hole. Applying the phase-stepping method with 7 images [28,29], the wrapped phases of the retardation and the principal direction are obtained as shown in Figs. 4(b) and (c). The phase map of the retardation in Fig. 4(b) contains the region of ambiguity where the mathematical sign of the retardation is wrong. In addition, the values of the retardation in Fig. 4(b) are lying in the range of the -n to n rad. On the other hand, the principal direction in Fig. 4(c) lies in the range from -n/4 to n/4 rad whereas the actual value should be in the range from -n/2 to n/2 rad. The ambiguity of the retardation is corrected using the phase maps obtained for the three monochromatic wavelengths as shown in Fig. 4(d). Then, the unwrapped phases of the retardation and the principal direction are obtained, as shown in Figs. 4(e) and (f).


 (a) Photoelastic fringe pattern; (b) wrapped retardation with ambiguity of sign; (c) wrapped principal direction; (c) wrapped retardation; (e) unwrapped retardation; (f) unwrapped principal direction

Fig. 4 (a) Photoelastic fringe pattern; (b) wrapped retardation with ambiguity of sign; (c) wrapped principal direction; (c) wrapped retardation; (e) unwrapped retardation; (f) unwrapped principal direction

Finite element model of the analysis region

Fig. 5 Finite element model of the analysis region

It is recognized that the absolute phase values at almost all points are obtained as shown in this figure. The principal stress difference and the principal direction at the number of the data points M = 2394 on the specimen surface are extracted and used as the data input into the algorithm by the proposed method.

The stress separation is performed in the 20 mm x 20 mm region around the hole, indicated by ABCD, shown in Fig. 3. Figure 5 shows the finite element model of the 20 mm x 20 mm region used for the proposed method. In this model, 8-noded isoparametric elements are used. The numbers of the elements and the nodes are 200 and 680, respectively. In order to obtain the stresses under the unit force at a point on the boundary, the displacements at some nodes must be fixed to prevent the rigid body motion. In this study, the x and y components of the displacement at the point A and the y directional displacement at the point B are assumed not to displace though these points are displaced actually. This assumption is valid because the rigid body translation and the rotation of the analysis region do not affect the stress distribution.

Example of the variation of nodal force during iteration process in nonlinear algorithm

Fig. 6 Example of the variation of nodal force during iteration process in nonlinear algorithm

Tractions along the boundary CD

Fig. 7 Tractions along the boundary CD

The nodal forces at the other nodes on the boundary are obtained by the proposed method. The number of the nodes along the boundary is 80 and thus the number of the nodal forces along the boundary is 160. That is, the number of the nodal forces to be determined is N = 157 because the three displacement components at the points A and B are fixed.

The nodal forces are determined using both linear and nonlinear algorithms. Figure 6 shows an example of the variation of nodal force at a point during iteration process in the nonlinear algorithm. In this example, the initial value of -1 N is given as shown in this figure. The value of the nodal force is corrected by the Newton-Raphson method. Then, the value is converged to a constant value as shown. Because the nodal forces at 157 points are simultaneously determined in the iteration process, the convergence is not fast and the number of iteration of about 40 is required in this example.

Stresses obtained by finite element direct analysis: (a) ax; (b) ay; (c) xxy

Fig. 10 Stresses obtained by finite element direct analysis: (a) ax; (b) ay; (c) xxy

The tractions along the boundary CD determined from the nodal forces obtained by the linear and nonlinear algorithms are shown in Fig. 7. In this figure, solid curves represent the values obtained by finite element direct analysis. As shown in this figure, the traction on the boundary of the analysis area obtained by the proposed method show good agreement with the values obtained by the direct analysis. In addition, it is observed that the accuracies of the tractions obtained by linear and nonlinear algorithms are almost same. Using the nodal forces obtained by the proposed method as the input data to finite element analysis, the stresses are computed. Figures 8 and 9 show the stresses around the hole obtained by the linear and nonlinear algorithm, respectively. The stresses obtained by finite element direct analysis are also shown in Fig. 10 for comparison. As shown in these figures, the stress components are obtained from the photoelastic fringes by the proposed linear and nonlinear algorithms. The average difference between the y directional normal stresses ay obtained by the linear algorithm and direct ones is 0.11 MPa, the maximum difference is 0.66 MPa, and the standard deviation is 0.10 MPa. On the other hand, the average difference, the maximum difference and the standard deviation between the values by the nonlinear algorithm and those by the direct analysis are 0.28 MPa, 0.68 MPa, and 0.12 MPa, respectively. It seems that the results obtained by the linear algorithm is better than those by the nonlinear algorithm. However, in linear algorithm, the principal direction as well as the principal stress difference is used for obtaining the shear stress and the normal stress difference. Therefore, the results of the stress separation are affected by the accuracy of the principal direction. The principal stress difference can be accurately evaluated in photoe-lasticity. As mentioned, however, it is known that the accurate evaluation of the principal direction is difficult even if a phase-stepping method is introduced. In this case, therefore, the nonlinear algorithm should be used for obtaining the better results. The drawback of the nonlinear algorithm is that the sign of the nodal force cannot be determined. Therefore, appropriate initial values of the nodal force is determined by the linear algorithm and then the nonlinear algorithm is used for determining the tractions if the accurate result of the principal direction is not obtained. Then, the stresses with appropriate sign can be obtained.

CONCLUSIONS

In this study, an experimental-numerical hybrid method for determining stress components in photoelasticity is proposed. Boundary conditions for a local finite element model are inversely determined from the principal stress difference and the principal direction in linear algorithm. On the other hand, the boundary conditions can be determined from the principal stress difference if the nonlinear algorithm is used. Then, the stresses are obtained by finite element direct analysis using the computed boundary conditions. The effectiveness of the proposed method is validated by analyzing the stresses around a hole in a perforated plate under tension. Results show that the boundary conditions of the local finite element model can be determined from the photoelastic fringes and then the stresses can be obtained by the proposed method.

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